# $p$-Euler equations and $p$-Navier-Stokes equations

## TL;DR

This paper introduces $p$-Euler and $p$-Navier-Stokes equations derived from Wasserstein-$p$ distances, exploring their structure, properties, and proving the global existence of weak solutions in certain cases.

## Contribution

The work formulates new $p$-Euler and $p$-Navier-Stokes equations based on optimal transport principles and establishes existence results for weak solutions in specific parameter regimes.

## Key findings

- $p$-Euler equations relate to Wasserstein-$p$ distances.
- Existence of weak solutions for $p$-Navier-Stokes in $
eal^d$ for $ extgamma=p$ and $p extgreater= d extgreater= 2.
- $p$-Laplacian appears in the vorticity formulation in 2D.

## Abstract

We propose in this work new systems of equations which we call $p$-Euler equations and $p$-Navier-Stokes equations. $p$-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-$p$ distances, with incompressibility constraint. $p$-Euler equations have similar structures with the usual Euler equations but the `momentum' is the signed ($p-1$)-th power of the velocity. In the 2D case, the $p$-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the $p$-Laplacian of the streamfunction. By adding diffusion presented by $\gamma$-Laplacian of the velocity, we obtain what we call $p$-Navier-Stokes equations. If $\gamma=p$, the {\it a priori} energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the $p$-Navier-Stokes equations in $\mathbb{R}^d$ for $\gamma=p$ and $p\ge d\ge 2$ through a compactness criterion.